The program I use to find the existing gaps (which are all reserved) also outputs a file that groups the n as follows:

Code:

14 <= n < 100, each group has 1 n
100 <= n < 1000, each group has 10 n from xx0 to xx9
1000 <= n < 10000, each group has 100 n from xx00 to xx99
10000 <= n < 100000, each group has 1000 n from xx000 to xx999

into a file called groups.txt. For each group it lists the max k that the group was tested to. You will notice that the file uses scientific notation to represent that k. This makes the data far more readable. Looking at the data implies that the size of the range of k should be a multiple of certain values determined by the n. For example:

- if n < 100, k should be a multiple of 1e14
- if 100 <= n < 399, k should be a multiple of 1e12
- if 400 <= n < 999, k should be a multiple of 1e11
- if 1000 <= n < 9999, k should be a multiple of 1e8
- if 10000 <= n < 39999, k should be a multiple of 1e6
- if 40000 <= n < 99999, k should be a multiple of 1e4

If you look at the groups.txt file in the attached zip file, you will see exactly how I came up with these numbers. Based upon this data I would add these to the "Most Wanted" ranges:

- n : 900-949 , k :180e to 200e9
- n : 950-999 , k :150e9 to 200e9
- n : 8000-8999 , k : 350e6 to 400e6
- n : 9000-9999 , k : 260e6 to 300e6
- n : 60000-61999 , k : 125e3 to 130e3

When I look at the file, I also see ranges like this:

Code:

n 4400-4499 maxk 2000e6
n 4500-4599 maxk 700e6
n 4600-4699 maxk 700e6
n 4700-4799 maxk 700e6
n 4800-4899 maxk 700e6
n 4900-4999 maxk 700e6
n 5000-5099 maxk 1200e6

Where subsequent ranges are tested to a higher k than previous ranges. I would add this to the "Most Wanted" ranges:

- n : 4500-4999 , k : 700e6 to 1200e6

If this is desirable information to mine from the FermatSearch pages, then I can modify the program to output these ranges in addition to the gaps.